LMFDB - Embedded newform 4200.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.5371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.47214 q^{13} +6.47214 q^{17} +2.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{27} -8.47214 q^{29} -0.472136 q^{31} +2.00000 q^{33} +2.47214 q^{37} +4.47214 q^{39} -3.52786 q^{41} -2.47214 q^{43} +6.47214 q^{47} +1.00000 q^{49} -6.47214 q^{51} +2.00000 q^{53} -2.00000 q^{57} -3.52786 q^{61} +1.00000 q^{63} +1.52786 q^{67} -4.00000 q^{69} +12.4721 q^{71} +7.52786 q^{73} -2.00000 q^{77} +8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} +8.47214 q^{87} -17.4164 q^{89} -4.47214 q^{91} +0.472136 q^{93} +3.52786 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{27} - 8 q^{29} + 8 q^{31} + 4 q^{33} - 4 q^{37} - 16 q^{41} + 4 q^{43} + 4 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 4 q^{57} - 16 q^{61} + 2 q^{63} + 12 q^{67} - 8 q^{69} + 16 q^{71} + 24 q^{73} - 4 q^{77} + 2 q^{81} + 8 q^{83} + 8 q^{87} - 8 q^{89} - 8 q^{93} + 16 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^27 - 8 * q^29 + 8 * q^31 + 4 * q^33 - 4 * q^37 - 16 * q^41 + 4 * q^43 + 4 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 - 4 * q^57 - 16 * q^61 + 2 * q^63 + 12 * q^67 - 8 * q^69 + 16 * q^71 + 24 * q^73 - 4 * q^77 + 2 * q^81 + 8 * q^83 + 8 * q^87 - 8 * q^89 - 8 * q^93 + 16 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.47214	0.406417	0.203208	−	0.979136i	$-0.434863\pi$
$37$	2.47214	0.406417	0.203208	+	0.979136i	$0.434863\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−3.52786	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$41$	−3.52786	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.47214	−0.906280
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.52786	0.186658	0.0933292	−	0.995635i	$-0.470249\pi$
$67$	1.52786	0.186658	0.0933292	+	0.995635i	$0.470249\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	12.4721	1.48017	0.740085	−	0.672513i	$-0.234785\pi$
$71$	12.4721	1.48017	0.740085	+	0.672513i	$0.234785\pi$
$72$	0	0
$73$	7.52786	0.881070	0.440535	−	0.897735i	$-0.354789\pi$
$73$	7.52786	0.881070	0.440535	+	0.897735i	$0.354789\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.94427	−0.542704	−0.271352	−	0.962480i	$-0.587471\pi$
$83$	−4.94427	−0.542704	−0.271352	+	0.962480i	$0.587471\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.47214	0.908308
$88$	0	0
$89$	−17.4164	−1.84614	−0.923068	−	0.384637i	$-0.874327\pi$
$89$	−17.4164	−1.84614	−0.923068	+	0.384637i	$0.874327\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	0.472136	0.0489582
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.52786	0.358200	0.179100	−	0.983831i	$-0.442681\pi$
$97$	3.52786	0.358200	0.179100	+	0.983831i	$0.442681\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	12.9443	1.27544	0.637719	−	0.770270i	$-0.279878\pi$
$103$	12.9443	1.27544	0.637719	+	0.770270i	$0.279878\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	−2.47214	−0.234645
$112$	0	0
$113$	14.9443	1.40584	0.702919	−	0.711269i	$-0.251879\pi$
$113$	14.9443	1.40584	0.702919	+	0.711269i	$0.251879\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	6.47214	0.593300
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	3.52786	0.318097
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	2.47214	0.217659
$130$	0	0
$131$	13.8885	1.21345	0.606724	−	0.794913i	$-0.292483\pi$
$131$	13.8885	1.21345	0.606724	+	0.794913i	$0.292483\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	0	0
$139$	−2.94427	−0.249730	−0.124865	−	0.992174i	$-0.539850\pi$
$139$	−2.94427	−0.249730	−0.124865	+	0.992174i	$0.539850\pi$
$140$	0	0
$141$	−6.47214	−0.545052
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.4721	−1.02176	−0.510879	−	0.859653i	$-0.670680\pi$
$149$	−12.4721	−1.02176	−0.510879	+	0.859653i	$0.670680\pi$
$150$	0	0
$151$	−17.8885	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$152$	0	0
$153$	6.47214	0.523241
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.41641	0.751511	0.375756	−	0.926719i	$-0.377383\pi$
$157$	9.41641	0.751511	0.375756	+	0.926719i	$0.377383\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	23.4164	1.83411	0.917057	−	0.398755i	$-0.130558\pi$
$163$	23.4164	1.83411	0.917057	+	0.398755i	$0.130558\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.4721	0.810358	0.405179	−	0.914237i	$-0.367209\pi$
$167$	10.4721	0.810358	0.405179	+	0.914237i	$0.367209\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.9443	−1.71494	−0.857468	−	0.514538i	$-0.827964\pi$
$179$	−22.9443	−1.71494	−0.857468	+	0.514538i	$0.827964\pi$
$180$	0	0
$181$	16.4721	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$181$	16.4721	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$182$	0	0
$183$	3.52786	0.260787
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−12.9443	−0.946579
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−17.4164	−1.26021	−0.630104	−	0.776511i	$-0.716988\pi$
$191$	−17.4164	−1.26021	−0.630104	+	0.776511i	$0.716988\pi$
$192$	0	0
$193$	12.9443	0.931749	0.465875	−	0.884851i	$-0.345740\pi$
$193$	12.9443	0.931749	0.465875	+	0.884851i	$0.345740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.8885	−1.41700	−0.708500	−	0.705711i	$-0.750628\pi$
$197$	−19.8885	−1.41700	−0.708500	+	0.705711i	$0.750628\pi$
$198$	0	0
$199$	20.4721	1.45123	0.725616	−	0.688100i	$-0.241555\pi$
$199$	20.4721	1.45123	0.725616	+	0.688100i	$0.241555\pi$
$200$	0	0
$201$	−1.52786	−0.107767
$202$	0	0
$203$	−8.47214	−0.594627
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	25.8885	1.78224	0.891120	−	0.453767i	$-0.149920\pi$
$211$	25.8885	1.78224	0.891120	+	0.453767i	$0.149920\pi$
$212$	0	0
$213$	−12.4721	−0.854577
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.472136	−0.0320507
$218$	0	0
$219$	−7.52786	−0.508686
$220$	0	0
$221$	−28.9443	−1.94700
$222$	0	0
$223$	20.9443	1.40253	0.701266	−	0.712900i	$-0.252618\pi$
$223$	20.9443	1.40253	0.701266	+	0.712900i	$0.252618\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.88854	−0.656326	−0.328163	−	0.944621i	$-0.606429\pi$
$227$	−9.88854	−0.656326	−0.328163	+	0.944621i	$0.606429\pi$
$228$	0	0
$229$	−26.3607	−1.74196	−0.870981	−	0.491316i	$-0.836516\pi$
$229$	−26.3607	−1.74196	−0.870981	+	0.491316i	$0.836516\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−2.94427	−0.192886	−0.0964428	−	0.995339i	$-0.530746\pi$
$233$	−2.94427	−0.192886	−0.0964428	+	0.995339i	$0.530746\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.94427	−0.580993
$238$	0	0
$239$	15.5279	1.00441	0.502207	−	0.864747i	$-0.332522\pi$
$239$	15.5279	1.00441	0.502207	+	0.864747i	$0.332522\pi$
$240$	0	0
$241$	−11.8885	−0.765808	−0.382904	−	0.923788i	$-0.625076\pi$
$241$	−11.8885	−0.765808	−0.382904	+	0.923788i	$0.625076\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.94427	−0.569110
$248$	0	0
$249$	4.94427	0.313331
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4164	1.46068	0.730338	−	0.683086i	$-0.239363\pi$
$257$	23.4164	1.46068	0.730338	+	0.683086i	$0.239363\pi$
$258$	0	0
$259$	2.47214	0.153611
$260$	0	0
$261$	−8.47214	−0.524412
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	17.4164	1.06587
$268$	0	0
$269$	30.3607	1.85112	0.925562	−	0.378597i	$-0.123593\pi$
$269$	30.3607	1.85112	0.925562	+	0.378597i	$0.123593\pi$
$270$	0	0
$271$	24.4721	1.48658	0.743288	−	0.668971i	$-0.233265\pi$
$271$	24.4721	1.48658	0.743288	+	0.668971i	$0.233265\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.4164	1.40696	0.703478	−	0.710717i	$-0.251629\pi$
$277$	23.4164	1.40696	0.703478	+	0.710717i	$0.251629\pi$
$278$	0	0
$279$	−0.472136	−0.0282660
$280$	0	0
$281$	2.94427	0.175641	0.0878203	−	0.996136i	$-0.472010\pi$
$281$	2.94427	0.175641	0.0878203	+	0.996136i	$0.472010\pi$
$282$	0	0
$283$	16.9443	1.00723	0.503616	−	0.863927i	$-0.332003\pi$
$283$	16.9443	1.00723	0.503616	+	0.863927i	$0.332003\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.52786	−0.208243
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	−3.52786	−0.206807
$292$	0	0
$293$	24.9443	1.45726	0.728630	−	0.684908i	$-0.240157\pi$
$293$	24.9443	1.45726	0.728630	+	0.684908i	$0.240157\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−17.8885	−1.03452
$300$	0	0
$301$	−2.47214	−0.142492
$302$	0	0
$303$	−4.47214	−0.256917
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−12.9443	−0.736374
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−25.4164	−1.43662	−0.718310	−	0.695723i	$-0.755084\pi$
$313$	−25.4164	−1.43662	−0.718310	+	0.695723i	$0.755084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.94427	0.390029	0.195015	−	0.980800i	$-0.437525\pi$
$317$	6.94427	0.390029	0.195015	+	0.980800i	$0.437525\pi$
$318$	0	0
$319$	16.9443	0.948697
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.9443	0.720239
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.9443	−0.826420
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	33.8885	1.86268	0.931341	−	0.364147i	$-0.118639\pi$
$331$	33.8885	1.86268	0.931341	+	0.364147i	$0.118639\pi$
$332$	0	0
$333$	2.47214	0.135472
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.0557	−0.602244	−0.301122	−	0.953586i	$-0.597361\pi$
$337$	−11.0557	−0.602244	−0.301122	+	0.953586i	$0.597361\pi$
$338$	0	0
$339$	−14.9443	−0.811661
$340$	0	0
$341$	0.944272	0.0511352
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.8328	1.65519	0.827596	−	0.561324i	$-0.189708\pi$
$347$	30.8328	1.65519	0.827596	+	0.561324i	$0.189708\pi$
$348$	0	0
$349$	25.4164	1.36051	0.680255	−	0.732976i	$-0.261869\pi$
$349$	25.4164	1.36051	0.680255	+	0.732976i	$0.261869\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	17.5279	0.932914	0.466457	−	0.884544i	$-0.345530\pi$
$353$	17.5279	0.932914	0.466457	+	0.884544i	$0.345530\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.47214	−0.342542
$358$	0	0
$359$	13.4164	0.708091	0.354045	−	0.935228i	$-0.384806\pi$
$359$	13.4164	0.708091	0.354045	+	0.935228i	$0.384806\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.8885	−1.35137	−0.675685	−	0.737190i	$-0.736152\pi$
$367$	−25.8885	−1.35137	−0.675685	+	0.737190i	$0.736152\pi$
$368$	0	0
$369$	−3.52786	−0.183653
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−16.3607	−0.847124	−0.423562	−	0.905867i	$-0.639220\pi$
$373$	−16.3607	−0.847124	−0.423562	+	0.905867i	$0.639220\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.8885	1.95136
$378$	0	0
$379$	−5.88854	−0.302474	−0.151237	−	0.988498i	$-0.548326\pi$
$379$	−5.88854	−0.302474	−0.151237	+	0.988498i	$0.548326\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−20.3607	−1.04038	−0.520191	−	0.854050i	$-0.674139\pi$
$383$	−20.3607	−1.04038	−0.520191	+	0.854050i	$0.674139\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.47214	−0.125666
$388$	0	0
$389$	−2.58359	−0.130993	−0.0654967	−	0.997853i	$-0.520863\pi$
$389$	−2.58359	−0.130993	−0.0654967	+	0.997853i	$0.520863\pi$
$390$	0	0
$391$	25.8885	1.30924
$392$	0	0
$393$	−13.8885	−0.700584
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.52786	−0.377813	−0.188906	−	0.981995i	$-0.560494\pi$
$397$	−7.52786	−0.377813	−0.188906	+	0.981995i	$0.560494\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−15.8885	−0.793436	−0.396718	−	0.917941i	$-0.629851\pi$
$401$	−15.8885	−0.793436	−0.396718	+	0.917941i	$0.629851\pi$
$402$	0	0
$403$	2.11146	0.105179
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.94427	−0.245078
$408$	0	0
$409$	−7.88854	−0.390063	−0.195032	−	0.980797i	$-0.562481\pi$
$409$	−7.88854	−0.390063	−0.195032	+	0.980797i	$0.562481\pi$
$410$	0	0
$411$	19.8885	0.981030
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.94427	0.144182
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−10.9443	−0.533391	−0.266696	−	0.963781i	$-0.585932\pi$
$421$	−10.9443	−0.533391	−0.266696	+	0.963781i	$0.585932\pi$
$422$	0	0
$423$	6.47214	0.314686
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.52786	−0.170725
$428$	0	0
$429$	−8.94427	−0.431834
$430$	0	0
$431$	−21.4164	−1.03159	−0.515796	−	0.856711i	$-0.672504\pi$
$431$	−21.4164	−1.03159	−0.515796	+	0.856711i	$0.672504\pi$
$432$	0	0
$433$	21.4164	1.02921	0.514603	−	0.857428i	$-0.327939\pi$
$433$	21.4164	1.02921	0.514603	+	0.857428i	$0.327939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	31.5279	1.50474	0.752371	−	0.658739i	$-0.228910\pi$
$439$	31.5279	1.50474	0.752371	+	0.658739i	$0.228910\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−36.9443	−1.75528	−0.877638	−	0.479325i	$-0.840882\pi$
$443$	−36.9443	−1.75528	−0.877638	+	0.479325i	$0.840882\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.4721	0.589912
$448$	0	0
$449$	−9.05573	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$449$	−9.05573	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$450$	0	0
$451$	7.05573	0.332241
$452$	0	0
$453$	17.8885	0.840477
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.94427	−0.418395	−0.209198	−	0.977873i	$-0.567085\pi$
$457$	−8.94427	−0.418395	−0.209198	+	0.977873i	$0.567085\pi$
$458$	0	0
$459$	−6.47214	−0.302093
$460$	0	0
$461$	−12.4721	−0.580885	−0.290443	−	0.956892i	$-0.593803\pi$
$461$	−12.4721	−0.580885	−0.290443	+	0.956892i	$0.593803\pi$
$462$	0	0
$463$	−34.8328	−1.61882	−0.809409	−	0.587245i	$-0.800212\pi$
$463$	−34.8328	−1.61882	−0.809409	+	0.587245i	$0.800212\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.0557	−0.881794	−0.440897	−	0.897558i	$-0.645340\pi$
$467$	−19.0557	−0.881794	−0.440897	+	0.897558i	$0.645340\pi$
$468$	0	0
$469$	1.52786	0.0705502
$470$	0	0
$471$	−9.41641	−0.433885
$472$	0	0
$473$	4.94427	0.227338
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−11.0557	−0.504098
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.8885	0.629350	0.314675	−	0.949199i	$-0.398104\pi$
$487$	13.8885	0.629350	0.314675	+	0.949199i	$0.398104\pi$
$488$	0	0
$489$	−23.4164	−1.05893
$490$	0	0
$491$	1.05573	0.0476443	0.0238222	−	0.999716i	$-0.492416\pi$
$491$	1.05573	0.0476443	0.0238222	+	0.999716i	$0.492416\pi$
$492$	0	0
$493$	−54.8328	−2.46955
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.4721	0.559452
$498$	0	0
$499$	−5.88854	−0.263607	−0.131804	−	0.991276i	$-0.542077\pi$
$499$	−5.88854	−0.263607	−0.131804	+	0.991276i	$0.542077\pi$
$500$	0	0
$501$	−10.4721	−0.467861
$502$	0	0
$503$	−39.4164	−1.75749	−0.878745	−	0.477291i	$-0.841619\pi$
$503$	−39.4164	−1.75749	−0.878745	+	0.477291i	$0.841619\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	−10.5836	−0.469109	−0.234555	−	0.972103i	$-0.575363\pi$
$509$	−10.5836	−0.469109	−0.234555	+	0.972103i	$0.575363\pi$
$510$	0	0
$511$	7.52786	0.333013
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.9443	−0.569288
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−9.41641	−0.412540	−0.206270	−	0.978495i	$-0.566133\pi$
$521$	−9.41641	−0.412540	−0.206270	+	0.978495i	$0.566133\pi$
$522$	0	0
$523$	−7.05573	−0.308525	−0.154263	−	0.988030i	$-0.549300\pi$
$523$	−7.05573	−0.308525	−0.154263	+	0.988030i	$0.549300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.05573	−0.133110
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.7771	0.683382
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.9443	0.990118
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−22.9443	−0.986451	−0.493226	−	0.869901i	$-0.664182\pi$
$541$	−22.9443	−0.986451	−0.493226	+	0.869901i	$0.664182\pi$
$542$	0	0
$543$	−16.4721	−0.706887
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.3050	1.59504	0.797522	−	0.603289i	$-0.206144\pi$
$547$	37.3050	1.59504	0.797522	+	0.603289i	$0.206144\pi$
$548$	0	0
$549$	−3.52786	−0.150566
$550$	0	0
$551$	−16.9443	−0.721850
$552$	0	0
$553$	8.94427	0.380349
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.05573	0.214218	0.107109	−	0.994247i	$-0.465841\pi$
$557$	5.05573	0.214218	0.107109	+	0.994247i	$0.465841\pi$
$558$	0	0
$559$	11.0557	0.467607
$560$	0	0
$561$	12.9443	0.546508
$562$	0	0
$563$	34.8328	1.46803	0.734014	−	0.679134i	$-0.237645\pi$
$563$	34.8328	1.46803	0.734014	+	0.679134i	$0.237645\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−10.1115	−0.423151	−0.211576	−	0.977362i	$-0.567859\pi$
$571$	−10.1115	−0.423151	−0.211576	+	0.977362i	$0.567859\pi$
$572$	0	0
$573$	17.4164	0.727581
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.4721	1.01879	0.509394	−	0.860533i	$-0.329870\pi$
$577$	24.4721	1.01879	0.509394	+	0.860533i	$0.329870\pi$
$578$	0	0
$579$	−12.9443	−0.537946
$580$	0	0
$581$	−4.94427	−0.205123
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.0557	−0.621416	−0.310708	−	0.950505i	$-0.600566\pi$
$587$	−15.0557	−0.621416	−0.310708	+	0.950505i	$0.600566\pi$
$588$	0	0
$589$	−0.944272	−0.0389080
$590$	0	0
$591$	19.8885	0.818105
$592$	0	0
$593$	−21.3050	−0.874890	−0.437445	−	0.899245i	$-0.644116\pi$
$593$	−21.3050	−0.874890	−0.437445	+	0.899245i	$0.644116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.4721	−0.837869
$598$	0	0
$599$	18.3607	0.750197	0.375099	−	0.926985i	$-0.377609\pi$
$599$	18.3607	0.750197	0.375099	+	0.926985i	$0.377609\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	1.52786	0.0622194
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.94427	−0.200682	−0.100341	−	0.994953i	$-0.531993\pi$
$607$	−4.94427	−0.200682	−0.100341	+	0.994953i	$0.531993\pi$
$608$	0	0
$609$	8.47214	0.343308
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	−7.41641	−0.299546	−0.149773	−	0.988720i	$-0.547854\pi$
$613$	−7.41641	−0.299546	−0.149773	+	0.988720i	$0.547854\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−47.8885	−1.92792	−0.963960	−	0.266047i	$-0.914282\pi$
$617$	−47.8885	−1.92792	−0.963960	+	0.266047i	$0.914282\pi$
$618$	0	0
$619$	11.8885	0.477841	0.238920	−	0.971039i	$-0.423206\pi$
$619$	11.8885	0.477841	0.238920	+	0.971039i	$0.423206\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−17.4164	−0.697774
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	23.0557	0.917834	0.458917	−	0.888479i	$-0.348238\pi$
$631$	23.0557	0.917834	0.458917	+	0.888479i	$0.348238\pi$
$632$	0	0
$633$	−25.8885	−1.02898
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	12.4721	0.493390
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−7.05573	−0.278251	−0.139125	−	0.990275i	$-0.544429\pi$
$643$	−7.05573	−0.278251	−0.139125	+	0.990275i	$0.544429\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.2492	−1.50373	−0.751866	−	0.659316i	$-0.770846\pi$
$647$	−38.2492	−1.50373	−0.751866	+	0.659316i	$0.770846\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.472136	0.0185045
$652$	0	0
$653$	1.05573	0.0413138	0.0206569	−	0.999787i	$-0.493424\pi$
$653$	1.05573	0.0413138	0.0206569	+	0.999787i	$0.493424\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.52786	0.293690
$658$	0	0
$659$	−12.1115	−0.471795	−0.235898	−	0.971778i	$-0.575803\pi$
$659$	−12.1115	−0.471795	−0.235898	+	0.971778i	$0.575803\pi$
$660$	0	0
$661$	48.2492	1.87668	0.938339	−	0.345717i	$-0.112364\pi$
$661$	48.2492	1.87668	0.938339	+	0.345717i	$0.112364\pi$
$662$	0	0
$663$	28.9443	1.12410
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.8885	−1.31217
$668$	0	0
$669$	−20.9443	−0.809752
$670$	0	0
$671$	7.05573	0.272383
$672$	0	0
$673$	21.8885	0.843741	0.421871	−	0.906656i	$-0.361374\pi$
$673$	21.8885	0.843741	0.421871	+	0.906656i	$0.361374\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.11146	0.0811499	0.0405749	−	0.999176i	$-0.487081\pi$
$677$	2.11146	0.0811499	0.0405749	+	0.999176i	$0.487081\pi$
$678$	0	0
$679$	3.52786	0.135387
$680$	0	0
$681$	9.88854	0.378930
$682$	0	0
$683$	−14.1115	−0.539960	−0.269980	−	0.962866i	$-0.587017\pi$
$683$	−14.1115	−0.539960	−0.269980	+	0.962866i	$0.587017\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.3607	1.00572
$688$	0	0
$689$	−8.94427	−0.340750
$690$	0	0
$691$	−48.8328	−1.85769	−0.928844	−	0.370471i	$-0.879196\pi$
$691$	−48.8328	−1.85769	−0.928844	+	0.370471i	$0.879196\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.8328	−0.864855
$698$	0	0
$699$	2.94427	0.111363
$700$	0	0
$701$	−0.472136	−0.0178323	−0.00891616	−	0.999960i	$-0.502838\pi$
$701$	−0.472136	−0.0178323	−0.00891616	+	0.999960i	$0.502838\pi$
$702$	0	0
$703$	4.94427	0.186477
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.47214	0.168192
$708$	0	0
$709$	35.8885	1.34782	0.673911	−	0.738812i	$-0.264613\pi$
$709$	35.8885	1.34782	0.673911	+	0.738812i	$0.264613\pi$
$710$	0	0
$711$	8.94427	0.335436
$712$	0	0
$713$	−1.88854	−0.0707265
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.5279	−0.579899
$718$	0	0
$719$	8.94427	0.333565	0.166783	−	0.985994i	$-0.446662\pi$
$719$	8.94427	0.333565	0.166783	+	0.985994i	$0.446662\pi$
$720$	0	0
$721$	12.9443	0.482070
$722$	0	0
$723$	11.8885	0.442140
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.0557	−0.706738	−0.353369	−	0.935484i	$-0.614964\pi$
$727$	−19.0557	−0.706738	−0.353369	+	0.935484i	$0.614964\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−5.63932	−0.208293	−0.104147	−	0.994562i	$-0.533211\pi$
$733$	−5.63932	−0.208293	−0.104147	+	0.994562i	$0.533211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.05573	−0.112559
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	8.94427	0.328576
$742$	0	0
$743$	2.83282	0.103926	0.0519630	−	0.998649i	$-0.483452\pi$
$743$	2.83282	0.103926	0.0519630	+	0.998649i	$0.483452\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.94427	−0.180901
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.4721	−1.10753	−0.553764	−	0.832673i	$-0.686809\pi$
$757$	−30.4721	−1.10753	−0.553764	+	0.832673i	$0.686809\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−24.4721	−0.887114	−0.443557	−	0.896246i	$-0.646284\pi$
$761$	−24.4721	−0.887114	−0.443557	+	0.896246i	$0.646284\pi$
$762$	0	0
$763$	14.9443	0.541019
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1.05573	0.0380705	0.0190353	−	0.999819i	$-0.493941\pi$
$769$	1.05573	0.0380705	0.0190353	+	0.999819i	$0.493941\pi$
$770$	0	0
$771$	−23.4164	−0.843321
$772$	0	0
$773$	−16.9443	−0.609443	−0.304722	−	0.952441i	$-0.598563\pi$
$773$	−16.9443	−0.609443	−0.304722	+	0.952441i	$0.598563\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.47214	−0.0886874
$778$	0	0
$779$	−7.05573	−0.252798
$780$	0	0
$781$	−24.9443	−0.892576
$782$	0	0
$783$	8.47214	0.302769
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	14.9443	0.531357
$792$	0	0
$793$	15.7771	0.560261
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	41.8885	1.48191
$800$	0	0
$801$	−17.4164	−0.615379
$802$	0	0
$803$	−15.0557	−0.531305
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.3607	−1.06875
$808$	0	0
$809$	−24.8328	−0.873075	−0.436538	−	0.899686i	$-0.643795\pi$
$809$	−24.8328	−0.873075	−0.436538	+	0.899686i	$0.643795\pi$
$810$	0	0
$811$	13.0557	0.458449	0.229224	−	0.973374i	$-0.426381\pi$
$811$	13.0557	0.458449	0.229224	+	0.973374i	$0.426381\pi$
$812$	0	0
$813$	−24.4721	−0.858275
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.94427	−0.172978
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	19.3050	0.673747	0.336874	−	0.941550i	$-0.390630\pi$
$821$	19.3050	0.673747	0.336874	+	0.941550i	$0.390630\pi$
$822$	0	0
$823$	15.0557	0.524810	0.262405	−	0.964958i	$-0.415484\pi$
$823$	15.0557	0.524810	0.262405	+	0.964958i	$0.415484\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.7214	−1.13783	−0.568917	−	0.822395i	$-0.692637\pi$
$827$	−32.7214	−1.13783	−0.568917	+	0.822395i	$0.692637\pi$
$828$	0	0
$829$	3.52786	0.122528	0.0612639	−	0.998122i	$-0.480487\pi$
$829$	3.52786	0.122528	0.0612639	+	0.998122i	$0.480487\pi$
$830$	0	0
$831$	−23.4164	−0.812306
$832$	0	0
$833$	6.47214	0.224246
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.472136	0.0163194
$838$	0	0
$839$	32.9443	1.13736	0.568681	−	0.822558i	$-0.307454\pi$
$839$	32.9443	1.13736	0.568681	+	0.822558i	$0.307454\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	−2.94427	−0.101406
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−16.9443	−0.581526
$850$	0	0
$851$	9.88854	0.338975
$852$	0	0
$853$	−31.5279	−1.07949	−0.539747	−	0.841827i	$-0.681480\pi$
$853$	−31.5279	−1.07949	−0.539747	+	0.841827i	$0.681480\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.5836	−0.566485	−0.283242	−	0.959048i	$-0.591410\pi$
$857$	−16.5836	−0.566485	−0.283242	+	0.959048i	$0.591410\pi$
$858$	0	0
$859$	−21.0557	−0.718412	−0.359206	−	0.933258i	$-0.616952\pi$
$859$	−21.0557	−0.718412	−0.359206	+	0.933258i	$0.616952\pi$
$860$	0	0
$861$	3.52786	0.120229
$862$	0	0
$863$	18.8328	0.641077	0.320538	−	0.947236i	$-0.396136\pi$
$863$	18.8328	0.641077	0.320538	+	0.947236i	$0.396136\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−24.8885	−0.845259
$868$	0	0
$869$	−17.8885	−0.606827
$870$	0	0
$871$	−6.83282	−0.231521
$872$	0	0
$873$	3.52786	0.119400
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.3050	1.66491	0.832455	−	0.554093i	$-0.186935\pi$
$877$	49.3050	1.66491	0.832455	+	0.554093i	$0.186935\pi$
$878$	0	0
$879$	−24.9443	−0.841349
$880$	0	0
$881$	27.5279	0.927437	0.463719	−	0.885983i	$-0.346515\pi$
$881$	27.5279	0.927437	0.463719	+	0.885983i	$0.346515\pi$
$882$	0	0
$883$	−16.3607	−0.550581	−0.275290	−	0.961361i	$-0.588774\pi$
$883$	−16.3607	−0.550581	−0.275290	+	0.961361i	$0.588774\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.58359	−0.153902	−0.0769510	−	0.997035i	$-0.524518\pi$
$887$	−4.58359	−0.153902	−0.0769510	+	0.997035i	$0.524518\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	12.9443	0.433164
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.8885	0.597281
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	12.9443	0.431236
$902$	0	0
$903$	2.47214	0.0822675
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.3607	−0.543247	−0.271624	−	0.962404i	$-0.587561\pi$
$907$	−16.3607	−0.543247	−0.271624	+	0.962404i	$0.587561\pi$
$908$	0	0
$909$	4.47214	0.148331
$910$	0	0
$911$	−25.4164	−0.842083	−0.421042	−	0.907041i	$-0.638335\pi$
$911$	−25.4164	−0.842083	−0.421042	+	0.907041i	$0.638335\pi$
$912$	0	0
$913$	9.88854	0.327263
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8885	0.458640
$918$	0	0
$919$	−36.7214	−1.21133	−0.605663	−	0.795721i	$-0.707092\pi$
$919$	−36.7214	−1.21133	−0.605663	+	0.795721i	$0.707092\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−55.7771	−1.83593
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.9443	0.425146
$928$	0	0
$929$	19.3050	0.633375	0.316687	−	0.948530i	$-0.397429\pi$
$929$	19.3050	0.633375	0.316687	+	0.948530i	$0.397429\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.3050	−0.499991	−0.249995	−	0.968247i	$-0.580429\pi$
$937$	−15.3050	−0.499991	−0.249995	+	0.968247i	$0.580429\pi$
$938$	0	0
$939$	25.4164	0.829433
$940$	0	0
$941$	35.5279	1.15818	0.579088	−	0.815265i	$-0.303409\pi$
$941$	35.5279	1.15818	0.579088	+	0.815265i	$0.303409\pi$
$942$	0	0
$943$	−14.1115	−0.459532
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.0000	−0.519930	−0.259965	−	0.965618i	$-0.583711\pi$
$947$	−16.0000	−0.519930	−0.259965	+	0.965618i	$0.583711\pi$
$948$	0	0
$949$	−33.6656	−1.09283
$950$	0	0
$951$	−6.94427	−0.225183
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−16.9443	−0.547731
$958$	0	0
$959$	−19.8885	−0.642235
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	0	0
$969$	−12.9443	−0.415830
$970$	0	0
$971$	−33.8885	−1.08754	−0.543768	−	0.839236i	$-0.683003\pi$
$971$	−33.8885	−1.08754	−0.543768	+	0.839236i	$0.683003\pi$
$972$	0	0
$973$	−2.94427	−0.0943890
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.8328	−1.69027	−0.845136	−	0.534552i	$-0.820480\pi$
$977$	−52.8328	−1.69027	−0.845136	+	0.534552i	$0.820480\pi$
$978$	0	0
$979$	34.8328	1.11326
$980$	0	0
$981$	14.9443	0.477134
$982$	0	0
$983$	−25.5279	−0.814212	−0.407106	−	0.913381i	$-0.633462\pi$
$983$	−25.5279	−0.814212	−0.407106	+	0.913381i	$0.633462\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.47214	−0.206010
$988$	0	0
$989$	−9.88854	−0.314437
$990$	0	0
$991$	−56.9443	−1.80889	−0.904447	−	0.426586i	$-0.859716\pi$
$991$	−56.9443	−1.80889	−0.904447	+	0.426586i	$0.859716\pi$
$992$	0	0
$993$	−33.8885	−1.07542
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.63932	−0.178599	−0.0892995	−	0.996005i	$-0.528463\pi$
$997$	−5.63932	−0.178599	−0.0892995	+	0.996005i	$0.528463\pi$
$998$	0	0
$999$	−2.47214	−0.0782149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bj.1.1		2
4.3	odd	2		8400.2.a.db.1.1		2
5.2	odd	4		840.2.t.c.169.4	yes	4
5.3	odd	4		840.2.t.c.169.2	✓	4
5.4	even	2		4200.2.a.bk.1.2		2
15.2	even	4		2520.2.t.f.1009.2		4
15.8	even	4		2520.2.t.f.1009.1		4
20.3	even	4		1680.2.t.h.1009.4		4
20.7	even	4		1680.2.t.h.1009.2		4
20.19	odd	2		8400.2.a.cz.1.2		2
60.23	odd	4		5040.2.t.u.1009.2		4
60.47	odd	4		5040.2.t.u.1009.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.c.169.2	✓	4	5.3	odd	4
840.2.t.c.169.4	yes	4	5.2	odd	4
1680.2.t.h.1009.2		4	20.7	even	4
1680.2.t.h.1009.4		4	20.3	even	4
2520.2.t.f.1009.1		4	15.8	even	4
2520.2.t.f.1009.2		4	15.2	even	4
4200.2.a.bj.1.1		2	1.1	even	1	trivial
4200.2.a.bk.1.2		2	5.4	even	2
5040.2.t.u.1009.1		4	60.47	odd	4
5040.2.t.u.1009.2		4	60.23	odd	4
8400.2.a.cz.1.2		2	20.19	odd	2
8400.2.a.db.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.47214 q^{13} +6.47214 q^{17} +2.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{27} -8.47214 q^{29} -0.472136 q^{31} +2.00000 q^{33} +2.47214 q^{37} +4.47214 q^{39} -3.52786 q^{41} -2.47214 q^{43} +6.47214 q^{47} +1.00000 q^{49} -6.47214 q^{51} +2.00000 q^{53} -2.00000 q^{57} -3.52786 q^{61} +1.00000 q^{63} +1.52786 q^{67} -4.00000 q^{69} +12.4721 q^{71} +7.52786 q^{73} -2.00000 q^{77} +8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} +8.47214 q^{87} -17.4164 q^{89} -4.47214 q^{91} +0.472136 q^{93} +3.52786 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{27} - 8 q^{29} + 8 q^{31} + 4 q^{33} - 4 q^{37} - 16 q^{41} + 4 q^{43} + 4 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 4 q^{57} - 16 q^{61} + 2 q^{63} + 12 q^{67} - 8 q^{69} + 16 q^{71} + 24 q^{73} - 4 q^{77} + 2 q^{81} + 8 q^{83} + 8 q^{87} - 8 q^{89} - 8 q^{93} + 16 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^27 - 8 * q^29 + 8 * q^31 + 4 * q^33 - 4 * q^37 - 16 * q^41 + 4 * q^43 + 4 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 - 4 * q^57 - 16 * q^61 + 2 * q^63 + 12 * q^67 - 8 * q^69 + 16 * q^71 + 24 * q^73 - 4 * q^77 + 2 * q^81 + 8 * q^83 + 8 * q^87 - 8 * q^89 - 8 * q^93 + 16 * q^97 - 4 * q^99

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